GRAPHS AND COMMON FIXED POINT THEOREMS FOR SEQUENCES OF MAPS

We demonstrate a usefulness of the notion of a connected graph for obtaining some 
common fixed point theorems. In particular, we establish two theorems of this type involving one, 
two and four sequences of maps. This generalizes among others the recent results of S. Chang [1], 
J. Jachymski [2], S. Sessa, R. N. Mukherjee and T. Som [3], and T. Taniguchi [4].

There are a number of papers dealing with fixed points for sequences of maps.B. E. Rhoades [5] divided these results into four categories (for the details, see [5], p. 10).In this paper we extend the fourth class of theorems discussed in [5].In this situation maps A and Aj or A and Bj satisfy pairwise a contractive type condition involving eventually other maps, with contractive gauge functions, which may depend on and j.To get a common fixed point theorem, most of the authors use an iteration procedure involving all of the maps considered (see, e.g., [3], [4], [6]).
Then, however, the same contractive gauge function is employed, or some additional hypothesis on gauge functions are imposed (see, e.g., the comments in [5], p. 13-14).Recently, G. Jungck et al. [7] and B. E. Rhoades [5] obtained common fixed point theorems for a sequence of maps using earlier results involving a finite number of maps.As has been pointed out in [5], such a way of treatment enables one to use different contractive gauge functions, without any additional hypotheses (see Corollary 1 [5]).A theorem of this type has been also established in the recent article [2].
In this paper we give a precise description of a set of positive integers (i,j) for which a contractive condition is to be satisfied in order to guarantee the existence of a common fixed point.Most of the authors assume that a contractive condition is to hold for all (i, j) with j (see, e.g., [1], [3], [7], [8]).However, H. Chatterji has observed that it suffices to use a contractive conditions for pairs (i,j) e J :--{(1,n + 1) "n e N} only (N denotes the set of all positive integers).The same set J is employed in Theorem 5.1 [2].
All of the theorems mentioned here deal with a single sequence of maps.On the other hand, T.
Taniguchi [4] establishing a fixed point theorem for two sequences of maps, has assumed that a contractive condition holds for all pairs (i,j) E JT := {(2n 1,2n) n e N} u {2n,2m / 1)-m >_ n _> (1.1) In the next section we show that, thanks to the notion of a connected graph, it is possible to unify and extend all of the above results.Moreover, our conditions imposed on a set J involving a graph appear to be necessary and sufficient for the existence of a common fixed point (see Theorems 2.2 and 2.3).

A FIXED POINT THEOREM INVOLVING GRAPH.
We start by recalling a common fixed point theorem for four maps ([2], Theorem 3.3).It can be deduced from Lemma 1 [9], that though this theorem involves a contractive gauge function, it yields the recent result of Jungck et al. [7] involving (e, 5)-type conditions.For the definition of compatible maps, a generalization of the commutative map concept, see [10].The letter R+ denotes the set of all nonegative reals.THEOREM 2.1.Let A, B, S and T be selfmaps of a complete metric space (X, d), and let :R+ R+ be an upper sernicontinuous (not necessarily monotonic) ]unction such that () < t for t > O. Let (A,B,S, T) satisfy he following conditions: d(Ax, By) <_ V(maz{d(Sz, Ty), d(Az, Sx), d(By, Ty), ( ][d(Ax, Ty) + d(By, Sx)]}), ]or a/l x, y X; A(X) C_ T(X) and B(X) C_ S(X); (2.2) the pairs A, S and B, T are compatible; (2.3) one of A, B, S and T is continuous. (2.4) Then A, B, S and T have a unique common fzed point.
Farther, let us recall that an undirected graph is a pair < V, E >, where V is a set and E is a family of two-element subsets of V. A graph < V, E > is said to be connected if for each x, y e V, there exists a finite sequence {xi}'_--0 such that x0 x, x, y and {xi-l,x} E E, for 1, n (see, e.g., [11]).The letters II1 and II2 denote the projections of N onto N, i.e., II1(n, m) := n, and II2(n, m) m, for (n, m) e N.
THEOREM 2.2.Let A, B, S, T and A,, B,, S, (n N) be selfmaps of a complete metric space (X, d).Let J C_ N and for (i,j) J, {,i lt+ It+ be an upper sernicontinuous function such that {ii(t) < t for t > O. Assume that on__g of the following conditions hold.
(2) All the A (n N), S and T if (2.6) holds.
(4) All the A, (n N), B and S if (2.9) holds.
PROOF.Assume that one of conditions (2.5)-(2.9)holds.By Theorem 2.1, for each (i,j) J there is a unique common fixed point zdi for the suitable quaternion of maps.Let (i,j), (k,l) J and {i,j} f {k,l} .Assume that j k or j 1.Then, by putting in (2.1), x := zii and y := zz, and replacing (A, B, S, T) by the suitable quaternion of maps, we obtain that d(x,y) < i(d(x,y)).Hence, x y since I, ii(t) < t for t > 0. Assume that k or 1.Similarly, by putting in (2.1), x := zkt and y := zii we obtain that d(x, y) <_ Therefore, zii zt.Now, let (i,j) and (k,l) be arbitrary elements of J.By the connectivity of < J, Ej >, there exists an n N and a sequence {(i,J)}=0 in J such that (io,jo) (i,j), (i.,j) (k,l), and {ik-l,j-a} f {i,j} , for k 1 ,n.Then, by the preceding part of the proof, for k 1, n, which immediately gives zi zt.This means, there is a z0 X such that zii zo, for all (i,j) J. Now, fix an n E N and assume that (2.5) holds.By hypothesis, there is a k E N such that (k,n) J or (n,k) J.For example, assume (k,n) J. Then z is the common fixed point of A, B, Sk and Sn; in particular, z0(= z) is the common fixed point of .4,B and Sn.The same argument may be used in the cases, in which any one of conditions (2.6)-(2.9)holds instead of (2.5).
(4) REMARK 2.2.By putting J :--J1 and assigning the i# to be the same function, and assuming that (2.5) or (2.6) holds, Theorem 2.2 yields Theorem 2 of Chang [1] and Theorem 3 of Sessa et al. [3], respectively.Clearly, we may also put here J := Jk for any k e {2, 3, 4} in order to obtain the essential extensions of the above theorems.REMARK 2.3.By putting in Theorem 2.2, J := J2 and @i1 := kt for some k e (0,1), t e 1 and (i,j) e J, and assuming that (2.7) holds, we obtain the result generalizing Theorem A of Taniguchi [4] who has employed the condition d(Ax, Ajy) <_ kd(Bz, Byy), for x, y e X and (i, j) Jr (Jr is defined by (1.1)), which is less general than (2.1).
REMARK 2.5.Clearly, conditions (2.5)-(2.9)may be weakened.It suffices to have the suit- able quaternions of maps satisfy the assumptions of Theorem 2.1.We have slightly strengthened them for aesthetic reasons.
The following theorem is a converse to Theorem 2.2.It appears that the connectivity of the graph < J, Ej > and the condition IIa(J) UIIz(J) N are necessary for the existence of a common fixed point.More precisely, we have.
THEOREM 3.1.Let J C_ N and Ej be defined as in Theorem 2.2.If the graph < J, Ej > is not connected or Ha(J)U II(J) N, then there e.zist a complete metric space (X,d) and selymaps S, T, An (n N) of X for which (2.6) is satisfied with ij being the same linear function, and there is no common jed point for the yamay oy maps.
PROOF.Assume that I/(J)U II(J) N. Then there exists an no 1N such that, for all k N, (k, no) J and (n0, k) J. Let X := R+ be endowed with the Euclidean metric, S := I, the identity on X, T := S, Ax:=x for n:no and xEX, A0x:=x+l (xX),